Whitney’s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S2 can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term “twisting” and “2-switching” respectively. With the twisting operation, we give a pure geometrical proof of Whitney’s 2-switching theorem. As an application, we obtain some relationships between two knots which correspond to the same signed planar graph. Besides, we also give a necessary and sufficient condition to test whether a pair of reduced alternating diagrams are mutants of each other by their signed planar graphs.

In this paper we use the T1 theorem to prove a new characterization with minimum regularity and cancellation conditions for inhomogeneous Besov and Triebel-Lizorkin spaces over spaces of homogeneous type. These results are new even for ℝn.

The Mehler formula for the Hermite polynomials allows for an integral representation of the one–dimensional Fractional Fourier transform. In this paper, we introduce a multi–dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler’s formula for the generalized Clifford–Hermite polynomials of Clifford analysis.

Let σ = {σi|i ∈ I} be some partition of the set of all primes ℙ, G a finite group and σ(G) = {σi|σi ∩ π(G) ≠ ∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-semipermutable in G with respect to H if HHix
= HixH for all x ∈ G and all Hi ∈ H such that (|H|, |Hi|) = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.

In this paper, we discuss the Schatten-p class (0 < p ≤ ∞) of Toeplitz operators on generalized Fock space with the symbol in positive Borel measure. It makes a great difference from other papers by using the estimates of the kernel and the weight together instead of separately estimating each other. We also get the equivalent conditions when a Toeplitz operator is in the Schatten-p class.

In this paper, we will use the Schwarz lemma at the boundary to character the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized locally biholomorphic quasi-convex mappings in the unit ball Bn respectively.

In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the objective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces (Z, d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain X ⊂ Z is countably compact in any Hausdorff topology weaker than that induced by d. When (Z, d) is a Féchet space (i.e., a complete metrizable locally convex space), our existence result only requires that the domain X ⊂ Z is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems, which extend and improve the related known results.